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Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
A thirdorder bounded arithmetic theory for PSPACE
 of Lecture Notes in Computer Science
, 2004
"... Abstract. We present a novel thirdorder theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that ..."
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Cited by 7 (3 self)
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Abstract. We present a novel thirdorder theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that from any configuration in the game of Hex, at least one player has a winning strategy. We then exhibit a translation of theorems of W 1 1 into families of propositional tautologies with polynomialsize proofs in BPLK (a recent propositional proof system for PSPACE and an alternative to G). This translation is clearer and more natural in several respects than the analogous ones for previous PSPACE theories. Keywords: Bounded arithmetic, propositional proof complexity, PSPACE, quantified propositional calculus 1
Propositional PSPACE reasoning with Boolean programs versus quantified Boolean formulas
 In ICALP
, 2004
"... Abstract. We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new sys ..."
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Cited by 6 (4 self)
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Abstract. We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new system, BPLK, is polynomially equivalent to the system G, which is based on the familiar but very different quantified Boolean formula (QBF) characterization of PSPACE due to Stockmeyer and Meyer. This equivalence is proved by way of two translations, one of which uses an idea reminiscent of the ɛterms of Hilbert and Bernays. 1
Theories and Proof Systems for PSPACE and the EXPTime Hierarchy
, 2006
"... This dissertation concerns theories of bounded arithmetic and propositional proof systems associated with PSPACE and classes from the exponentialtime hierarchy. The secondorder viewpoint of Zambella and Cook associates secondorder theories of bounded arithmetic with various complexity classes by ..."
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Cited by 2 (2 self)
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This dissertation concerns theories of bounded arithmetic and propositional proof systems associated with PSPACE and classes from the exponentialtime hierarchy. The secondorder viewpoint of Zambella and Cook associates secondorder theories of bounded arithmetic with various complexity classes by studying the definable functions of strings, rather than numbers. This approach simplifies presentation of the theories and their propositional translations, and furthermore is applicable to complexity classes that previously had no corresponding theories. We adapt this viewpoint to large complexity classes from the exponentialtime hierarchy by adding a third sort, intended to represent exponentially long strings (“superstrings”), and capable of coding, for example, the computation of an exponentialtime Turing machine. Specifically, our main theories W i 1 and T W i 1 are associated with PSPACEΣp i−1 and EXPΣp i−1, respectively. We also develop a model for computation in this thirdorder setting including a function calculus, and define thirdorder analogues of ordinary complexity classes. We then obtain recursiontheoretic characterizations of our function classes for FP, FPSPACE and FEXP. We use our characterization of FPSPACE as the basis for an open theory for PSPACE that is a
Bounded Arithmetic vs. Propositional Proof Systems vs. Complexity Classes
, 2005
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, 2005
"... The broad relevance and importance of bounded arithmetic and propositional proof complexity are well appreciated; these subjects are two of three which are interconnected in ..."
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The broad relevance and importance of bounded arithmetic and propositional proof complexity are well appreciated; these subjects are two of three which are interconnected in
Encoding Nested Boolean Functions as Quantified Boolean Formulas
"... In this paper, we consider the problem of compactly representing nested instantiations of propositional subformulas with different arguments as quantified Boolean formulas (QBF). We develop a generic QBF encoding pattern which combines and generalizes existing QBF encoding techniques for simpler typ ..."
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In this paper, we consider the problem of compactly representing nested instantiations of propositional subformulas with different arguments as quantified Boolean formulas (QBF). We develop a generic QBF encoding pattern which combines and generalizes existing QBF encoding techniques for simpler types of redundancy. We obtain an equivalencepreserving transformation in linear time from the PSPACEcomplete language of nested Boolean functions (NBF), also called Boolean programs, to prenex QBF. A transformation in the other direction from QBF to NBF is also possible in at most quadratic time by simulating quantifier expansion.